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<HR>

<H3>variable command 
</H3>
<P><B>Syntax:</B>
</P>
<PRE>variable name style args ... 
</PRE>
<UL><LI>name = name of variable to define 

<LI>style = <I>delete</I> or <I>index</I> or <I>loop</I> or <I>world</I> or <I>universe</I> or <I>uloop</I> or <I>string</I> or <I>format</I> or <I>getenv</I> or <I>file</I> or <I>internal</I> or <I>equal</I> or <I>particle</I> or <I>grid</I> 

<PRE>  <I>delete</I> = no args
  <I>index</I> args = one or more strings
  <I>loop</I> args = N
    N = integer size of loop, loop from 1 to N inclusive
  <I>loop</I> args = N pad
    N = integer size of loop, loop from 1 to N inclusive
    pad = all values will be same length, e.g. 001, 002, ..., 100
  <I>loop</I> args = N1 N2
    N1,N2 = loop from N1 to N2 inclusive
  <I>loop</I> args = N1 N2 pad
    N1,N2 = loop from N1 to N2 inclusive
    pad = all values will be same length, e.g. 050, 051, ..., 100
  <I>world</I> args = one string for each partition of processors
  <I>universe</I> args = one or more strings
  <I>uloop</I> args = N
    N = integer size of loop
  <I>uloop</I> args = N pad
    N = integer size of loop
    pad = all values will be same length, e.g. 001, 002, ..., 100
  <I>string</I> arg = one string
  <I>format</I> args = vname fstr
    vname = name of equal-style variable to evaluate
    fstr = C-style format string
  <I>getenv</I> arg = one string
  <I>file</I> arg = filename
  <I>internal</I> arg = numeric value
  <I>equal</I> or <I>particle</I> or <I>grid</I> args = one formula containing numbers, stats keywords, math operations, particle vectors, grid vectors, compute/fix/variable references
    numbers = 0.0, 100, -5.4, 2.8e-4, etc
    constants = PI
    stats keywords = step, np, vol, etc from <A HREF = "stats_style.html">stats_style</A>
    math operators = (), -x, x+y, x-y, x*y, x/y, x^y, x%y,
                     x==y, x!=y, x<y, x<=y, x>y, x>=y, x&&y, x||y, !x
    math functions = sqrt(x), exp(x), ln(x), log(x), abs(x),
                     sin(x), cos(x), tan(x), asin(x), acos(x), atan(x), atan2(y,x),
                     random(x,y), normal(x,y), ceil(x), floor(x), round(x)
                     ramp(x,y), stagger(x,y), logfreq(x,y,z), stride(x,y,z), vdisplace(x,y), swiggle(x,y,z), cwiggle(x,y,z)
    special functions = sum(x), min(x), max(x), ave(x), trap(x), slope(x), next(x)
    particle vectors = mass, type, mu, x, y, z, vx, vy, vz
    grid vectors = xc, yc, zc
    compute references = c_ID, c_ID[i], c_ID[i][j]
    fix references = f_ID, f_ID[i], f_ID[i][j]
    surface collision model references = s_ID[i]
    surface reaction model references = r_ID[i]
    variable references = v_name 
</PRE>

</UL>
<P><B>Examples:</B>
</P>
<PRE>variable x index run1 run2 run3 run4 run5 run6 run7 run8
variable LoopVar loop $n
variable beta equal temp/3.0
variable beta equal "temp / 3.0"
variable b equal c_myTemp
variable b particle x*y/vol
variable foo string myfile
variable foo internal 3.5
variable f file values.txt
variable temp world 300.0 310.0 320.0 ${Tfinal}
variable x universe 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
variable x uloop 15 pad
variable str format x %.6g
variable x delete 
</PRE>
<P><B>Description:</B>
</P>
<P>This command assigns one or more strings to a variable name for
evaluation later in the input script or during a simulation.
</P>
<P>Variables can thus be useful in several contexts.  A variable can be
defined and then referenced elsewhere in an input script to become
part of a new input command.  For variable styles that store multiple
strings, the <A HREF = "next.html">next</A> command can be used to increment which
string is assigned to the variable.  Variables of style <I>equal</I> store
a formula which when evaluated produces a single numeric value which
can be output either directly (see the <A HREF = "print.html">print</A>, <A HREF = "fix_print.html">fix
print</A>, and <A HREF = "run.html">run every</A> commands) or as part
of statistical output (see the <A HREF = "stats_style.html">stats_style</A>
command), or used as input to an averaging fix (see the <A HREF = "fix_ave_time.html">fix
ave/time</A> command).  Variables of style <I>particle</I>
store a formula which when evaluated produces one numeric value per
particle which can be output to a dump file (see the <A HREF = "dump.html">dump
particle</A> command).  Variables of style <I>internal</I> are used
by a few commands which set their value directly.
</P>
<P>In the discussion that follows, the "name" of the variable is the
arbitrary string that is the 1st argument in the variable command.
This name can only contain alphanumeric characters and underscores.
The "string" is one or more of the subsequent arguments.  The "string"
can be simple text as in the 1st example above, it can contain other
variables as in the 2nd example, or it can be a formula as in the 3rd
example.  The "value" is the numeric quantity resulting from
evaluation of the string.  Note that the same string can generate
different values when it is evaluated at different times during a
simulation.
</P>
<P>IMPORTANT NOTE: When an input script line is encountered that defines
a variable of style <I>equal</I> or <I>particle</I> or <I>grid</I> that contains a
formula, the formula is NOT immediately evaluated and the result
stored.  See the discussion below about "Immediate Evaluation of
Variables" if you want to do this.  This is also true of the <I>format</I>
style variable since it evaluates another variable when it is invoked.
</P>
<P>Variables of style <I>equal</I> and <I>particle</I> and <I>grid</I> can be used as
inputs to various other commands which evaluate their formulas as
needed, e.g. at different timesteps during a <A HREF = "run.html">run</A>.
</P>
<P>Variables of style <I>internal</I> can be used in place of an equal-style
variable, except by commands that set the value stored by the
internal-style variable.  Thus any command that states it can use an
equal-style variable as an argument, can also use an internal-style
variable.  This means that when the command evaluates the variable, it
will use the value set (internally) by another command.
</P>
<P>IMPORTANT NOTE: When a variable command is encountered in the input
script and the variable name has already been specified, the command
is ignored.  This means variables can NOT be re-defined in an input
script (with 2 exceptions, read further).  This is to allow an input
script to be processed multiple times without resetting the variables;
see the <A HREF = "jump.html">jump</A> or <A HREF = "include.html">include</A> commands.  It also
means that using the <A HREF = "Section_start.html#start_6">command-line switch</A>
-var will override a corresponding index variable setting in the input
script.
</P>
<P>There are two exceptions to this rule.  First, variables of style
<I>string</I>, <I>getenv</I>, <I>internal</I>, <I>equal</I>, and <I>particle</I> ARE redefined
each time the command is encountered.  This allows these style of
variables to be redefined multiple times in an input script.  In a
loop, this means the formula associated with an <I>equal</I> or <I>particle</I>
style variable can change if it contains a substitution for another
variable, e.g. $x or v_x.
</P>
<P>Second, as described below, if a variable is iterated on to the end of
its list of strings via the <A HREF = "next.html">next</A> command, it is removed
from the list of active variables, and is thus available to be
re-defined in a subsequent variable command.  The <I>delete</I> style does
the same thing.
</P>
<HR>

<P><A HREF = "Section_commands.html#cmd_2">Section 3.2</A> of the manual explains how
occurrences of a variable name in an input script line are replaced by
the variable's string.  The variable name can be referenced as $x if
the name "x" is a single character, or as ${LoopVar} if the name
"LoopVar" is one or more characters.
</P>
<P>As described below, for variable styles <I>index</I>, <I>loop</I>, <I>universe</I>,
and <I>uloop</I>, which string is assigned to a variable can be incremented
via the <A HREF = "next.html">next</A> command.  When there are no more strings to
assign, the variable is exhausted and a flag is set that causes the
next <A HREF = "jump.html">jump</A> command encountered in the input script to be
skipped.  This enables the construction of simple loops in the input
script that are iterated over and then exited from.
</P>
<P>As explained above, an exhausted variable can be re-used in an input
script.  The <I>delete</I> style also removes the variable, the same as if
it were exhausted, allowing it to be redefined later in the input
script or when the input script is looped over.  This can be useful
when breaking out of a loop via the <A HREF = "if.html">if</A> and <A HREF = "jump.html">jump</A>
commands before the variable would become exhausted.  For example,
</P>
<PRE>label	    loop
variable    a loop 5
print	    "A = $a"
if	    "$a > 2" then "jump in.script break"
next	    a
jump	    in.script loop
label	    break
variable    a delete 
</PRE>
<HR>

<P>This section describes how various variable styles are defined and
what they store.  Many of the styles store one or more strings.  Note
that a single string can contain spaces (multiple words), if it is
enclosed in quotes in the variable command.  When the variable is
substituted for in another input script command, its returned string
will then be interpreted as multiple arguments in the expanded
command.
</P>
<P>For the <I>index</I> style, one or more strings are specified.  Initially,
the 1st string is assigned to the variable.  Each time a
<A HREF = "next.html">next</A> command is used with the variable name, the next
string is assigned.  All processors assign the same string to the
variable.
</P>
<P><I>Index</I> style variables with a single string value can also be set by
using the command-line switch -var; see <A HREF = "Section_start.html#start_6">Section
2.6</A> of the manual for details.
</P>
<P>The <I>loop</I> style is identical to the <I>index</I> style except that the
strings are the integers from 1 to N inclusive, if only one argument N
is specified.  This allows generation of a long list of runs
(e.g. 1000) without having to list N strings in the input script.
Initially, the string "1" is assigned to the variable.  Each time a
<A HREF = "next.html">next</A> command is used with the variable name, the next
string ("2", "3", etc) is assigned.  All processors assign the same
string to the variable.  The <I>loop</I> style can also be specified with
two arguments N1 and N2.  In this case the loop runs from N1 to N2
inclusive, and the string N1 is initially assigned to the variable.
N1 <= N2 and N2 >= 0 is required.
</P>
<P>For the <I>world</I> style, one or more strings are specified.  There must
be one string for each processor partition or "world".  See <A HREF = "Section_start.html#start_6">Section
2.6</A> of the manual for information on
running SPARTA with multiple partitions via the "-partition"
command-line switch.  This variable command assigns one string to each
world.  All processors in the world are assigned the same string.  The
next command cannot be used with <I>equal</I> style variables, since there
is only one value per world.  This style of variable is useful when
you wish to run different simulations on different partitions.
</P>
<P>For the <I>universe</I> style, one or more strings are specified.  There
must be at least as many strings as there are processor partitions or
"worlds".  See <A HREF = "Section_start.html#start_6">this page</A> for information
on running SPARTA with multiple partitions via the "-partition"
command-line switch.  This variable command initially assigns one
string to each world.  When a <A HREF = "next.html">next</A> command is encountered
using this variable, the first processor partition to encounter it, is
assigned the next available string.  This continues until all the
variable strings are consumed.  Thus, this command can be used to run
50 simulations on 8 processor partitions.  The simulations will be run
one after the other on whatever partition becomes available, until
they are all finished.  <I>Universe</I> style variables are incremented
using the files "tmp.sparta.variable" and "tmp.sparta.variable.lock"
which you will see in your directory during such a SPARTA run.
</P>
<P>The <I>uloop</I> style is identical to the <I>universe</I> style except that the
strings are the integers from 1 to N.  This allows generation of long
list of runs (e.g. 1000) without having to list N strings in the input
script.
</P>
<P>For the <I>string</I> style, a single string is assigned to the variable.
The only difference between this and using the <I>index</I> style with a
single string is that a variable with <I>string</I> style can be redefined.
E.g. by another command later in the input script, or if the script is
read again in a loop.
</P>
<P>For the <I>format</I> style, an equal-style variable is specified along
with a C-style format string, e.g. "%f" or "%.10g", which must be
appropriate for formatting a double-precision floating-point value.
This allows an equal-style variable to be formatted specifically for
output as a string, e.g. by the <A HREF = "print.html">print</A> command, if the
default format "%.15g" has too much precision.
</P>
<P>For the <I>getenv</I> style, a single string is assigned to the variable
which should be the name of an environment variable.  When the
variable is evaluated, it returns the value of the environment
variable, or an empty string if it not defined.  This style of
variable can be used to adapt the behavior of SPARTA input scripts via
environment variable settings, or to retrieve information that has
been previously stored with the <A HREF = "shell.html">shell putenv</A> command.
Note that because environment variable settings are stored by the
operating systems, they persist beyond a <A HREF = "clear.html">clear</A> command.
</P>
<P>For the <I>file</I> style, a filename is provided which contains a list of
strings to assign to the variable, one per line.  The strings can be
numeric values if desired.  See the discussion of the next() function
below for equal-style variables, which will convert the string of a
file-style variable into a numeric value in a formula.
</P>
<P>When a file-style variable is defined, the file is opened and the
string on the first line is read and stored with the variable.  This
means the variable can then be evaluated as many times as desired and
will return that string.  There are two ways to cause the next string
from the file to be read: use the <A HREF = "next.html">next</A> command or the
next() function in an equal- or particle- or grid-style variable, as
discussed below.
</P>
<P>The rules for formatting the file are as follows.  A comment character
"#" can be used anywhere on a line; text starting with the comment
character is stripped.  Blank lines are skipped.  The first "word" of
a non-blank line, delimited by white space, is the "string" assigned
to the variable.
</P>
<P>For the <I>internal</I> style a numeric value is provided.  This value will
be assigned to the variable until a SPARTA command sets it to a new
value.  There is currently only one command that requirew <I>internal</I>
variables as inputs, because it resets them:
<A HREF = "create_particles.html">create_particles</A>.  As mentioned above, an
internal-style variable can be used in place of an equal-style
variable anywhere else in an input script, e.g. as an argument to
another command that allows for equal-style variables.
</P>
<HR>

<P>For the <I>equal</I> and <I>particle</I> and <I>grid</I> styles, a single string is
specified which represents a formula that will be evaluated afresh
each time the variable is used.  If you want spaces in the string,
enclose it in double quotes so the parser will treat it as a single
argument.  For <I>equal</I> style variables the formula computes a scalar
quantity, which becomes the value of the variable whenever it is
evaluated.  For <I>particle</I> style variables the formula computes one
quantity for each particle whenever it is evaluated.  For <I>grid</I> style
variables the formula computes one quantity for each grid cell
whenever it is evaluated.  A <I>grid</I> style variable computes quantites
for all flavors of child grid cells in the simulation, which includes
unsplit, cut, split, and sub cells.  See <A HREF = "Section_howto.html#howto_8">Section
4.8</A> of the manual gives details of how
SPARTA defines child, unsplit, split, and sub cells.
</P>
<P>Note that <I>equal</I> and <I>particle</I> and <I>grid</I> variables can produce
different values at different stages of the input script or at
different times during a run.  For example, if an <I>equal</I> variable is
used in a <A HREF = "fix_print.html">fix print</A> command, different values could
be printed each timestep it was invoked.  If you want a variable to be
evaluated immediately, so that the result is stored by the variable
instead of the string, see the section below on "Immediate Evaluation
of Variables".
</P>
<P>The next command cannot be used with <I>equal</I> or <I>particle</I> or <I>grid</I>
style variables, since there is only one string.
</P>
<P>The formula for an <I>equal</I> or <I>particle</I> or <I>grid</I> variable can
contain a variety of quantities.  The syntax for each kind of quantity
is simple, but multiple quantities can be nested and combined in
various ways to build up formulas of arbitrary complexity.  For
example, this is a valid (though strange) variable formula:
</P>
<PRE>variable x equal "np + c_MyTemp / vol^(1/3)" 
</PRE>
<P>Specifically, a formula can contain numbers, stats keywords, math
operators, math functions, particle vectors, grid vectors, compute
references, fix references, and references to other variables.
</P>
<DIV ALIGN=center><TABLE  BORDER=1 >
<TR><TD >Number</TD><TD > 0.2, 100, 1.0e20, -15.4, etc</TD></TR>
<TR><TD >Constant</TD><TD > PI</TD></TR>
<TR><TD >Stats keywords</TD><TD > step, np, vol, etc</TD></TR>
<TR><TD >Math operators</TD><TD > (), -x, x+y, x-y, x*y, x/y, x^y, x%y, x==y, x!=y, x<y, x<=y, x>y, x>=y, x&&y, x||y, !x</TD></TR>
<TR><TD >Math functions</TD><TD > sqrt(x), exp(x), ln(x), log(x), abs(x), sin(x), cos(x), tan(x),      asin(x), acos(x), atan(x), atan2(y,x), random(x,y,z), normal(x,y,z),      ceil(x), floor(x), round(x), ramp(x,y), stagger(x,y), logfreq(x,y,z),      stride(x,y,z), vdisplace(x,y), swiggle(x,y,z), cwiggle(x,y,z)</TD></TR>
<TR><TD >Special functions</TD><TD > sum(x), min(x), max(x), ave(x), trap(x), slope(x), next(x)</TD></TR>
<TR><TD >Particle vectors</TD><TD > mass, type, mu, x, y, z, vx, vy, vz</TD></TR>
<TR><TD >Grid vectors</TD><TD > xc, yc, zc</TD></TR>
<TR><TD >Compute references</TD><TD > c_ID, c_ID[i], c_ID[i][j]</TD></TR>
<TR><TD >Fix references</TD><TD > f_ID, f_ID[i], f_ID[i][j]</TD></TR>
<TR><TD >Surface collision model references</TD><TD > s_ID[i]</TD></TR>
<TR><TD >Surface reaction model references</TD><TD > r_ID[i]</TD></TR>
<TR><TD >Other variables</TD><TD > v_name 
</TD></TR></TABLE></DIV>

<HR>

<P>Most of the formula elements produce a scalar value.  A few produce a
per-particle vector or per-grid vector of values.  These are the
particle vectors or grid vectors, compute references that represent a
per-particle or per-grid vector, fix references that represent a
per-particle or per-grid vector, and variables that are particle-style
or grid-style variables.  Math functions that operate on scalar values
produce a scalar value; math function that operate on per-particle
vectors do so element-by-element and produce a per-particle vector.
</P>
<P>A formula for equal-style variables cannot use any formula element
that produces a per-particle or per-grid vector.  A formula for a
particle-style variable can use formula elements that produce either a
scalar value or a per-particle vector, but not a per-grid vector.
Likewise a grid-style variable can use formula elements that produce
either a scalar value or a per-grid vector, but not a per-particle
vector.
</P>
<P>The stats keywords allowed in a formula are those defined by the
<A HREF = "stats_style.html">stats_style custom</A> command.  If a variable is
evaluated directly in an input script (not during a run), then the
values accessed by the stats keyword must be current.  See the
discussion below about "Variable Accuracy".
</P>
<HR>

<H4>Math Operators 
</H4>
<P>Math operators are written in the usual way, where the "x" and "y" in
the examples can themselves be arbitrarily complex formulas, as in the
examples above.  In this syntax, "x" and "y" can be scalar values or
per-particle or per-grid vectors.  For example, "vol/np" is the
division of two scalars, where "vy+vz" is the element-by-element sum
of two per-particle vectors of y and z velocities.
</P>
<P>Operators are evaluated left to right and have the usual C-style
precedence: unary minus and unary logical NOT operator "!" have the
highest precedence, exponentiation "^" is next; multiplication and
division and the modulo operator "%" are next; addition and
subtraction are next; the 4 relational operators "<", "<=", ">", and
">=" are next; the two remaining relational operators "==" and "!="
are next; then the logical AND operator "&&"; and finally the logical
OR operator "||" has the lowest precedence.  Parenthesis can be used
to group one or more portions of a formula and/or enforce a different
order of evaluation than what would occur with the default precedence.
</P>
<P>IMPORTANT NOTE: Because a unary minus is higher precedence than
exponentiation, the formula "-2^2" will evaluate to 4, not -4.  This
convention is compatible with some programming languages, but not
others.  As mentioned, this behavior can be easily overridden with
parenthesis; the formula "-(2^2)" will evaluate to -4.
</P>
<P>The 6 relational operators return either a 1.0 or 0.0 depending on
whether the relationship between x and y is TRUE or FALSE.  For
example the expression x<10.0 in a particle-style variable formula
will return 1.0 for all particles whose x-coordinate is less than
10.0, and 0.0 for the others.  The logical AND operator will return
1.0 if both its arguments are non-zero, else it returns 0.0.  The
logical OR operator will return 1.0 if either of its arguments is
non-zero, else it returns 0.0.  The logical NOT operator returns 1.0
if its argument is 0.0, else it returns 0.0.
</P>
<P>These relational and logical operators can be used as a masking or
selection operation in a formula.  For example, the number of
particles whose properties satifsy one or more criteria could be
calculated by taking the returned per-particle vector of ones and
zeroes and passing it to the <A HREF = "compute_reduce.html">compute reduce</A>
command.
</P>
<HR>

<H4>Math Functions 
</H4>
<P>Math functions are specified as keywords followed by one or more
parenthesized arguments "x", "y", "z", each of which can themselves be
arbitrarily complex formulas.  In this syntax, the arguments can
represent scalar values or per-particle or per-grid vectors.  In the
latter cases, the math operation is performed on each element of the
vector.  For example, "sqrt(np)" is the sqrt() of a scalar, where
"sqrt(y*z)" yields a per-particle vector with each element being the
sqrt() of the product of one particle's y and z coordinates.
</P>
<P>Most of the math functions perform obvious operations.  The ln() is
the natural log; log() is the base 10 log.
</P>
<P>The random(x,y) function takes 2 arguments: x = lo and y = hi.  It
generates a uniform random number between lo and hi.  The normal(x,y)
function also takes 2 arguments: x = mu and y = sigma.  It generates a
Gaussian variate centered on mu with variance sigma^2.  For
equal-style variables, every processor uses the same random number
seed so that they each generate the same sequence of random numbers.
For particle-style or grid-style variables, a unique seed is created
for each processor.  This effectively generates a different random
number for each particle or grid cell being looped over in the
particle-style or grid-style variable.
</P>
<P>IMPORTANT NOTE: Internally, there is just one random number generator
for all equal-style variables and one for all particle-style and
grid-style variables.  If you define multiple variables (of each
style) which use the random() or normal() math functions, then the
internal random number generators will only be initialized once.
</P>
<P>The ceil(), floor(), and round() functions are those in the C math
library.  Ceil() is the smallest integer not less than its argument.
Floor() if the largest integer not greater than its argument.  Round()
is the nearest integer to its argument.
</P>
<P>The ramp(x,y) function uses the current timestep to generate a value
linearly intepolated between the specified x,y values over the course
of a run, according to this formula:
</P>
<PRE>value = x + (y-x) * (timestep-startstep) / (stopstep-startstep) 
</PRE>
<P>The run begins on startstep and ends on stopstep.  Startstep and
stopstep can span multiple runs, using the <I>start</I> and <I>stop</I> keywords
of the <A HREF = "run.html">run</A> command.  See the <A HREF = "run.html">run</A> command for
details of how to do this.
</P>
<P>IMPORTANT NOTE: Currently, the run command does not currently support
the start/stop keywords.  In the formula above startstep = 0 and
stopstep = the number of timesteps being performed by the run.
</P>
<P>The stagger(x,y) function uses the current timestep to generate a new
timestep.  X,y > 0 and x > y are required.  The generated timesteps
increase in a staggered fashion, as the sequence
x,x+y,2x,2x+y,3x,3x+y,etc.  For any current timestep, the next
timestep in the sequence is returned.  Thus if stagger(1000,100) is
used in a variable by the <A HREF = "dump_modify.html">dump_modify every</A>
command, it will generate the sequence of output timesteps:
</P>
<PRE>100,1000,1100,2000,2100,3000,etc 
</PRE>
<P>The logfreq(x,y,z) function uses the current timestep to generate a
new timestep.  X,y,z > 0 and y < z are required.  The generated
timesteps increase in a logarithmic fashion, as the sequence
x,2x,3x,...y*x,z*x,2*z*x,3*z*x,...y*z*x,z*z*x,2*z*x*x,etc.  For any
current timestep, the next timestep in the sequence is returned.  Thus
if logfreq(100,4,10) is used in a variable by the <A HREF = "dump_modify.html">dump_modify
every</A> command, it will generate the sequence of
output timesteps:
</P>
<PRE>100,200,300,400,1000,2000,3000,4000,10000,20000,etc 
</PRE>
<P>The stride(x,y,z) function uses the current timestep to generate a new
timestep.  X,y >= 0 and z > 0 and x <= y are required.  The generated
timesteps increase in increments of z, from x to y, I.e. it generates
the sequece x,x+z,x+2z,...,y.  If y-x is not a multiple of z, then
similar to the way a for loop operates, the last value will be one
that does not exceed y.  For any current timestep, the next timestep
in the sequence is returned.  Thus if stagger(1000,2000,100) is used
in a variable by the <A HREF = "dump_modify.html">dump_modify every</A> command, it
will generate the sequence of output timesteps:
</P>
<PRE>1000,1100,1200, ... ,1900,2000 
</PRE>
<P>The vdisplace(x,y) function takes 2 arguments: x = value0 and y =
velocity, and uses the elapsed time to change the value by a linear
displacement due to the applied velocity over the course of a run,
according to this formula:
</P>
<PRE>value = value0 + velocity*(timestep-startstep)*dt 
</PRE>
<P>where dt = the timestep size.
</P>
<P>The run begins on startstep.  Startstep can span multiple runs, using
the <I>start</I> keyword of the <A HREF = "run.html">run</A> command.  See the
<A HREF = "run.html">run</A> command for details of how to do this.  Note that the
<A HREF = "stats_style.html">stats_style</A> keyword <I>elaplong</I> =
timestep-startstep.
</P>
<P>The swiggle(x,y,z) and cwiggle(x,y,z) functions each take 3 arguments:
x = value0, y = amplitude, z = period.  They use the elapsed time to
oscillate the value by a sin() or cos() function over the course of a
run, according to one of these formulas, where omega = 2 PI / period:
</P>
<PRE>value = value0 + Amplitude * sin(omega*(timestep-startstep)*dt)
value = value0 + Amplitude * (1 - cos(omega*(timestep-startstep)*dt)) 
</PRE>
<P>where dt = the timestep size.
</P>
<P>The run begins on startstep.  Startstep can span multiple runs, using
the <I>start</I> keyword of the <A HREF = "run.html">run</A> command.  See the
<A HREF = "run.html">run</A> command for details of how to do this.  Note that the
<A HREF = "stats_style.html">stats_style</A> keyword <I>elaplong</I> =
timestep-startstep.
</P>
<HR>

<H4>Special Functions 
</H4>
<P>Special functions take specific kinds of arguments, meaning their
arguments cannot be formulas themselves.
</P>
<P>The sum(x), min(x), max(x), ave(x), trap(x), and slope(x) functions
each take 1 argument which is of the form "c_ID" or "c_ID[N]" or
"f_ID" or "f_ID[N]".  The first two are computes and the second two
are fixes; the ID in the reference should be replaced by the ID of a
compute or fix defined elsewhere in the input script.  The compute or
fix must produce either a global vector or array.  If it produces a
global vector, then the notation without "[N]" should be used.  If
it produces a global array, then the notation with "[N]" should be
used, when N is an integer, to specify which column of the global
array is being referenced.
</P>
<P>These functions operate on the global vector of inputs and reduce it
to a single scalar value.  This is analagous to the operation of the
<A HREF = "compute_reduce.html">compute reduce</A> command, which invokes the same
functions on per-particle or per-grid vectors.
</P>
<P>The sum() function calculates the sum of all the vector elements.  The
min() and max() functions find the minimum and maximum element
respectively.  The ave() function is the same as sum() except that it
divides the result by the length of the vector.  
</P>
<P>The trap() function is the same as sum() except the first and last
elements are multiplied by a weighting factor of 1/2 when performing
the sum.  This effectively implements an integratiion via the
trapezoidal rule on the global vector of data.  I.e. consider a set of
points, equally spaced by 1 in their x coordinate: (1,V1), (2,V2),
..., (N,VN), where the Vi are the values in the global vector of
length N.  The integral from 1 to N of these points is trap().
</P>
<P>The slope() function uses linear regression to fit a line to the set
of points, equally spaced by 1 in their x coordinate: (1,V1), (2,V2),
..., (N,VN), where the Vi are the values in the global vector of
length N.  The returned value is the slope of the line.  If the line
has a single point or is vertical, it returns 1.0e20.
</P>
<P>The next(x) function takes 1 argument which is a variable ID (not
"v_foo", just "foo").  It must be for a file-style
variable.  Each time the next() function is invoked (i.e. each time
the equal-style or particle-style or grid-style variable is evaluated),
the following steps occur.
</P>
<P>For file-style variables, the current string value stored by the
file-style variable is converted to a numeric value and returned by
the function.  And the next string value in the file is read and
stored.  Note that if the line previously read from the file was not a
numeric string, then it will typically evaluate to 0.0, which is
likely not what you want.
</P>
<P>Since file-style variables read and store the first line of the file
when they are defined in the input script, this is the value that will
be returned the first time the next() function is invoked.  If next()
is invoked more times than there are lines in the file, the variable
is deleted, similar to how the <A HREF = "next.html">next</A> command operates.
</P>
<HR>

<H4>Particle Vectors 
</H4>
<P>Particle vectors generate one value per particle, so that a reference
like "vx" means the x-component of each particles's velocity will be
used when evaluating the variable.  Some values are per-species
values, like mass and mu.  A reference like "mass" means the mass for
the particle's species.
</P>
<P>The meaning of the different particle vectors is self-explanatory.
</P>
<P>Particle vectors can only be used in <I>particle</I> style variables,
not in <I>equal</I> or <I>grid</I> style varaibles.
</P>
<HR>

<H4>Grid Vectors 
</H4>
<P>Particle vectors generate one value per grid cell, so that a reference
like "xc" means the x-component of the center point of the grid cell.
</P>
<P>The xc, yc, zc values are for the geometric center point of the grid
cell.
</P>
<P>Grid vectors can only be used in <I>grid</I> style variables, not in
<I>equal</I> or <I>particle</I> style varaibles.
</P>
<HR>

<H4>Compute References 
</H4>
<P>Compute references access quantities calculated by a
<A HREF = "compute.html">compute</A>.  The ID in the reference should be replaced by
the ID of a compute defined elsewhere in the input script.  As
discussed in the doc page for the <A HREF = "compute.html">compute</A> command,
computes can produce global, per-particle, per-grid, or per-surf
values.  Only global and per-particle and per-grid values can be used
in a variable.  Computes can also produce a scalar, vector, or array.
An equal-style variable can only use scalar values, which means a
global scalar, or an element of a global vector or array.
Particle-style variables can use the same scalar values.  They can
also use per-particle vector values.  A vector value can be a
per-particle vector itself, or a column of an per-particle array.
Grid-style variables can use the same scalar values.  They can also
use per-grid vector values.  A vector value can be a per-grid vector
itself, or a column of an per-grid array.  See the doc pages for
individual computes to see what kind of values they produce.
</P>
<P>Examples of different kinds of compute references are as follows.
There is no ambiguity as to what a reference means, since computes
only produce global or per-particle or per-grid quantities, never more
than one kind of quantity.
</P>
<DIV ALIGN=center><TABLE  BORDER=1 >
<TR><TD >c_ID</TD><TD > global scalar, or per-particle or per-grid vector</TD></TR>
<TR><TD >c_ID[I]</TD><TD > Ith element of global vector, or Ith column from per-particle or per-grid array</TD></TR>
<TR><TD >c_ID[I][J]</TD><TD > I,J element of global array 
</TD></TR></TABLE></DIV>

<P>For I and J, integers can be specified or a variable name, specified
as v_name, where name is the name of the variable, like
x[v_myIndex].  The variable can be of any style expect
particle-style.  The variable is evaluated and the result is expected
to be numeric and is cast to an integer (i.e. 3.4 becomes 3), to use
an an index, which must be a value from 1 to N.  Note that a "formula"
cannot be used as the argument between the brackets, e.g. x[243+10]
or x[v_myIndex+1] are not allowed.  To do this a single variable can
be defined that contains the needed formula.
</P>
<P>If a variable containing a compute is evaluated directly in an input
script (not during a run), then the values accessed by the compute
must be current.  See the discussion below about "Variable Accuracy".
</P>
<HR>

<H4>Fix References 
</H4>
<P>Fix references access quantities calculated by a <A HREF = "compute.html">fix</A>.
The ID in the reference should be replaced by the ID of a fix defined
elsewhere in the input script.  As discussed in the doc page for the
<A HREF = "fix.html">fix</A> command, fixes can produce global, per-particle,
per-grid, or per-surf values.  Only global and per-particle and
per-grid values can be used in a variable.  Fixes can also produce a
scalar, vector, or array.  An equal-style variable can only use scalar
values, which means a global scalar, or an element of a global vector
or array.  Particle-style variables can use the same scalar values.
They can also use per-particle vector values.  A vector value can be a
per-particle vector itself, or a column of an per-particle array.
Grid-style variables can use the same scalar values.  They can also
use per-grid vector values.  A vector value can be a per-grid vector
itself, or a column of an per-grid array.  See the doc pages for
individual fixes to see what kind of values they produce.
</P>
<P>The different kinds of fix references are exactly the same as the
compute references listed in the above table, where "c_" is replaced
by "f_".  Again, there is no ambiguity as to what a reference means,
since fixes only produce global or per-particle or per-grid
quantities, never more than one kind of quantity.
</P>
<DIV ALIGN=center><TABLE  BORDER=1 >
<TR><TD >f_ID</TD><TD > global scalar, or per-particle or per-grid vector</TD></TR>
<TR><TD >f_ID[I]</TD><TD > Ith element of global vector, or Ith column from per-particle or per-grid array</TD></TR>
<TR><TD >f_ID[I][J]</TD><TD > I,J element of global array 
</TD></TR></TABLE></DIV>

<P>For I and J, integers can be specified or a variable name, specified
as v_name, where name is the name of the variable.  The rules for this
syntax are the same as for the "Compute References" discussion above.
</P>
<P>If a variable containing a fix is evaluated directly in an input
script (not during a run), then the values accessed by the fix should
be current.  See the discussion below about "Variable Accuracy".
</P>
<P>Note that some fixes only generate quantities on certain timesteps.
If a variable attempts to access the fix on non-allowed timesteps, an
error is generated.  For example, the <A HREF = "fix_ave_time.html">fix ave/time</A>
command may only generate averaged quantities every 100 steps.  See
the doc pages for individual fix commands for details.
</P>
<HR>

<H4>Surface Collision and Surface Reaction Model References 
</H4>
<P>These references access quantities calculated by a
<A HREF = "surf_collide.html">surf_collide</A> or <A HREF = "surf_react.html">surf_react</A>
command.  The ID in the reference should be replaced by the ID of a
surface collision or surface reaction model defined elsewhere in the
input script.  As discussed in the doc pages for the
<A HREF = "surf_collide.html">surf_collide</A> and <A HREF = "surf_react.html">surf_react</A>
commands, these commmands produce global vectors, the elements of
which can be accessed by equal-style or particle-style or grid-style
variables, e.g.
</P>
<DIV ALIGN=center><TABLE  BORDER=1 >
<TR><TD >s_ID[I]</TD><TD > Ith element of global vector for a surface collision model</TD></TR>
<TR><TD >r_ID[I]</TD><TD > Ith element of global vector for a surface reaction model 
</TD></TR></TABLE></DIV>

<HR>

<H4>Variable References 
</H4>
<P>Variable references access quantities stored or calculated by other
variables, which will cause those variables to be evaluated.  The name
in the reference should be replaced by the name of a variable defined
elsewhere in the input script.
</P>
<P>As discussed on this doc page, equal-style variables generate a global
scalar numeric value; particle-style variables generate a per-particle
vector of numeric values; grid-style variables generate a per-grid
vector of numeric values; all other variables store a string.  The
formula for an equal-style variable can use any style of variable
except a particle- or grid-style.  The formula for a particle-style
variable can use any style of variable except a grid-style.  The
formula for a grid-style variable can use any style of variable except
a particle-style.  If a string-storing variable is used, the string is
converted to a numeric value.  Note that this will typically produce a
0.0 if the string is not a numeric string, which is likely not what
you want.  The formula for a particle-style variable can use any style
of variable, including other particle-style variables.
</P>
<P>Examples of different kinds of variable references are as follows.
There is no ambiguity as to what a reference means, since variables
produce only a global scalar or a per-particle or per-grid vector,
never more than one of these quantities.
</P>
<DIV ALIGN=center><TABLE  BORDER=1 >
<TR><TD >v_name</TD><TD > scalar, or per-particle or per-grid vector 
</TD></TR></TABLE></DIV>

<HR>

<P><B>Immediate Evaluation of Variables:</B>
</P>
<P>There is a difference between referencing a variable with a leading $
sign (e.g. $x or ${abc}) versus with a leading "v_" (e.g. v_x or
v_abc).  The former can be used in any input script command, including
a variable command.  The input script parser evaluates the reference
variable immediately and substitutes its value into the command.  As
explained in <A HREF = "Section_commands.html#3_2">Section commands 3.2</A> for
"Parsing rules", you can also use un-named "immediate" variables for
this purpose.  For example, a string like this
$((xlo+xhi)/2+sqrt(v_area)) in an input script command evaluates the
string between the parenthesis as an equal-style variable formula.
</P>
<P>Referencing a variable with a leading "v_" is an optional or required
kind of argument for some commands (e.g. the <A HREF = "fix_ave_spatial.html">fix
ave/spatial</A> or <A HREF = "dump.html">dump custom</A> or
<A HREF = "stats_style.html">stats_style</A> commands) if you wish it to evaluate
a variable periodically during a run.  It can also be used in a
variable formula if you wish to reference a second variable.  The
second variable will be evaluated whenever the first variable is
evaluated.
</P>
<P>As an example, suppose you use this command in your input script to
define the variable "n" as
</P>
<PRE>variable n equal np 
</PRE>
<P>before a run where the particle count changes.  You might think this
will assign the initial count to the variable "n".  That is not the
case.  Rather it assigns a formula which evaluates the count (using
the stats_style keyword "np") to the variable "n".  If you use the
variable "n" in some other command like <A HREF = "fix_ave_time.html">fix
ave/time</A> then the current particle count will be
evaluated continuously during the run.
</P>
<P>If you want to store the initial particle count of the system, it
can be done in this manner:
</P>
<PRE>variable n equal np
variable n0 equal $n 
</PRE>
<P>The second command will force "n" to be evaluated (yielding the
initial count) and assign that value to the variable "n0".  Thus the
command
</P>
<PRE>stats_style custom step v_n v_n0 
</PRE>
<P>would print out both the current and initial particle count
periodically during the run.
</P>
<P>Also note that it is a mistake to enclose a variable formula in quotes
if it contains variables preceeded by $ signs.  For example,
</P>
<PRE>variable nratio equal "${nfinal}/${n0}" 
</PRE>
<P>This is because the quotes prevent variable substitution (see <A HREF = "Section_commands.html#cmd_2">Section
2.2</A> of the manual on parsing input script
commands), and thus an error will occur when the formula for "nratio"
is evaluated later.
</P>
<HR>

<P><B>Variable Accuracy:</B>
</P>
<P>Obviously, SPARTA attempts to evaluate variables containing formulas
(<I>equal</I> and <I>particle</I> and <I>grid</I> style variables) accurately
whenever the evaluation is performed.  Depending on what is included
in the formula, this may require invoking a <A HREF = "compute.html">compute</A>, or
accessing a value previously calculated by a compute, or accessing a
value calculated and stored by a <A HREF = "fix.html">fix</A>.  If the compute is
one that calculates certain properties of the system such as the
pressure induced on a global boundary due to collisions, then these
quantities need to be tallied during the timesteps on which the
variable will need the values.
</P>
<P>SPARTA keeps track of all of this during a <A HREF = "run.html">run</A>.  An error
will be generated if you attempt to evaluate a variable on timesteps
when it cannot produce accurate values.  For example, if a
<A HREF = "stats_style.html">stats_style custom</A> command prints a variable which
accesses values stored by a <A HREF = "fix_ave_time.html">fix ave/time</A> command
and the timesteps on which stats output is generated are not multiples
of the averaging frequency used in the fix command, then an error will
occur.
</P>
<P>An input script can also request variables be evaluated before or
after or in between runs, e.g. by including them in a
<A HREF = "print.html">print</A> command.  In this case, if a compute is needed to
evaluate a variable (either directly or indirectly), SPARTA will not
invoke the compute, but it will use a value previously calculated by
the compute, and can do this only if it was invoked on the current
timestep.  Fixes will always provide a quantity needed by a variable,
but the quantity may or may not be current.  This leads to one of
three kinds of behavior:
</P>
<P>(1) The variable may be evaluated accurately.  If it contains
references to a compute or fix, and these values were calculated on
the last timestep of a preceeding run, then they will be accessed and
used by the variable and the result will be accurate.
</P>
<P>(2) SPARTA may not be able to evaluate the variable and will generate
an error message stating so.  For example, if the variable requires a
quantity from a <A HREF = "compute.html">compute</A> that has not been invoked on
the current timestep, SPARTA will generate an error.  This means, for
example, that such a variable cannot be evaluated before the first run
has occurred.  Likewise, in between runs, a variable containing a
compute cannot be evaluated unless the compute was invoked on the last
timestep of the preceding run, e.g. by stats output.
</P>
<P>One way to get around this problem is to perform a 0-timestep run
before using the variable.  For example, these commands
</P>
<PRE>compute myTemp grid all temp
variable t equal c_myTemp<B>1</B>
print "Initial temperature = $t"
run 1000 
</PRE>
<P>will generate an error if the run is the first run specified in the
input script, because generating a value for the "t" variable requires
a compute for calculating the temperature to be invoked.
</P>
<P>However, this sequence of commands would be fine:
</P>
<PRE>compute myTemp grid all temp
variable t equal c_myTemp<B>1</B>
run 0
print "Initial temperature = $t"
run 1000 
</PRE>
<P>The 0-timestep run initializes and invokes various computes, including
the one for temperature, so that the value it stores is current and
can be accessed by the variable "t" after the run has completed.  Note
that a 0-timestep run does not alter the state of the system, so it
does not change the input state for the 1000-timestep run that
follows.  Also note that the 0-timestep run must actually use and
invoke the compute in question (e.g. via <A HREF = "stats_style.html">stats</A> or
<A HREF = "dump.html">dump</A> output) in order for it to enable the compute to be
used in a variable after the run.  Thus if you are trying to print a
variable that uses a compute you have defined, you can insure it is
invoked on the last timestep of the preceding run by including it in
stats output.
</P>
<P>Unlike computes, <A HREF = "fix.html">fixes</A> will never generate an error if
their values are accessed by a variable in between runs.  They always
return some value to the variable.  However, the value may not be what
you expect if the fix has not yet calculated the quantity of interest
or it is not current.  For example, the <A HREF = "fix_indent.html">fix indent</A>
command stores the force on the indenter.  But this is not computed
until a run is performed.  Thus if a variable attempts to print this
value before the first run, zeroes will be output.  Again, performing
a 0-timestep run before printing the variable has the desired effect.
</P>
<P>(3) The variable may be evaluated incorrectly.  And SPARTA may have
no way to detect this has occurred.  Consider the following sequence
of commands:
</P>
<PRE>compute myTemp grid all temp
variable t equal c_myTemp<B>1</B>
run 1000
create_particles all n 10000
print "Final temperature = $t" 
</PRE>
<P>The first run is performed using the current set of particles.  The
temperature is evaluated on the final timestep and stored by the
<A HREF = "compute_grid.html">compute grid</A> compute (when invoked by the
<A HREF = "stats_style.html">stats_style</A> command).  Then new particles are added
by the <A HREF = "create_particles.html">create_particles</A> command, altering the
temperature of the system.  When the temperature is printed via the
"t" variable, SPARTA will use the temperature value stored by the
<A HREF = "compute_grid.html">compute grid</A> compute, thinking it is current.
There are many other commands which could alter the state of the
system between runs, causing a variable to evaluate incorrectly.
</P>
<P>The solution to this issue is the same as for case (2) above, namely
perform a 0-timestep run before the variable is evaluated to insure
the system is up-to-date.  For example, this sequence of commands
would print a temperature that reflected the new particles:
</P>
<PRE>compute myTemp grid all temp
variable t equal c_myTemp<B>1</B>
run 1000
create_particles all n 10000
run 0
print "Final temperature = $t" 
</PRE>
<HR>

<P><B>Restrictions:</B>
</P>
<P>All <I>universe</I>- and <I>uloop</I>-style variables defined in an input script
must have the same number of values.
</P>
<P><B>Related commands:</B>
</P>
<P><A HREF = "next.html">next</A>, <A HREF = "jump.html">jump</A>, <A HREF = "include.html">include</A>, <A HREF = "fix_print.html">fix
print</A>, <A HREF = "print.html">print</A>
</P>
<P><B>Default:</B> none
</P>
</HTML>
